CONDENSED MATTER PHYSICS

 A.A: 2025/26. Prof. Marco Grilli

Reception time
Tuesday 16-17 (by email appointment)
office 147 MARCONI blg

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The course starts on Wednesday 01/10/2025 in aula Amaldi Edificio Marconi (Marconi Blg.)

THE STUDENTS OF PROF. GRILLI'S CHANNEL ARE KINDLY REQUESTED TO SUBSCRIBE THE CLASSROOM PLATFORM OF THE COURSE FOR ANY FURTHER ANNOUNCEMENT

THE SUBSCRIPTION CODE IS: ccnyn6z 

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Lecture schedule:
Wednesday 8-10 (Aula Amaldi)
Thursday12-14 (Aula Cabibbo)
Friday 15-16 (Aula Cabibbo)

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Mid-term tests:
05/12/2025 17:00-19:00 (TBC) Aula Amaldi
16/01/2026 16:00-18:00 (TBC) Aula Amaldi

Exam dates:

***19/11/2025  at 16:00 written exam (Aula Rasetti, Marconi Blg)
***21/11/2025  at 09:00   (Prof. Grilli's office, room 147, Marconi Blg)

23/01/2026  at 09:00 written exam (Aula Amaldi, Marconi Blg)
29/01/2026  at  09:00 oral exam (Prof. Grilli's office, room 147, Marconi Blg)

03/02/2026 (notice that the date has changed)
at  09:00 extra oral exam (Prof. Grilli's office, room 147, Marconi Blg)
[only for QUARMEN and LA SCALA students]

17/02/2026  at 09:00 written exam (Aula Amaldi, Marconi Blg)
24/02/2026  at  09:00 oral exam (Prof. Grilli's office, room 147, Marconi Blg)

***22/05/2026  at 14:00 written exam (Aula Rasetti, Marconi Blg)
***25/05/2026 at 9:00   (Prof. Grilli's office, room 147, Marconi Blg)

15/06/2026  at 09:00 written exam (Aula Amaldi, Marconi Blg)
18/06/2026  at  09:00 oral exam (Prof. Grilli's office, room 147, Marconi Blg)

06/07/2026  at 09:00 written exam (Aula Amaldi, Marconi Blg)
09/07/2026  at  09:00 oral exam (Prof. Grilli's office, room 147, Marconi Blg) 

08/09/2025  at 09:00 written exam (Aula Amaldi, Marconi Blg)
10/09/2026  at  09:00 oral exam (Prof. Grilli's office, room 147, Marconi Blg)

*** This is a special session for some students only (please check the APPELLI-STRAORDINARI.pdf file on the Varie link to check if you are eligible)

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Program

- Crystal structures and Bravais lattice.
- Reciprocal lattice. Diffraction and solid crystals, structure factor.
- Electrons in solids, Bloch's theorem. Band structure. Tightly and weakly bound electrons. Holes and effective mass.
- Born-Oppheneimer approximation. Lattice vibrations, phonons, specific heat (Einstein's and Debye's model, density of states).
- Electrons in metals and interaction with an electromagnetic field (metal transport properties): Drude's and Sommerfeld's models. The semiclassical model.
- Intrinsic and extrinsic semiconductors. Temperature dependence of charge carrier density.
- Modern issues in transport: the Berry phase and the and the Anomalous Hall Effect.

Prerequisites
The course relies on the following prerequisites:

1. CLASSICAL MECHANICS reference text: H. Goldstein, C. P. Poole, and J. L. Safko Classical Mechanics, Addison-Wesley chapter 1 Survey of elementary principles - mechanics of a particle - mechanics of a system of particles - constraints - D'Alambert's principle and Lagrange's equations chapter 6 Oscillations - formulation of the problem - the eigenvalue equation and the principal axis transformation - frequencies of free vibration and normal coordinates chapter 8 The Hamilton equations of motion - Legendre transformations and the Hamilton equations of motion chapter 9 Canonical transformations - the equations of canonical transformations - Poisson brackets - Liouville's theorem

2. CLASSICAL ELECTROMAGNETISM reference text: D. Halliday, R. Resnick, and K. S. Crane Physics - part II, John Wiley & sons chapter 25 Electric charge and Coulomb's law - electric charge - conductors and insulators - Coulomb's law - continuous charge distributions - conservation of charge chapter 26 The electric field - the electric field - the electric field of point charges - the electric field of continuous charge distributions chapter 27 Gauss' law - the flux of the electric field - Gauss' law chapter 28 Electric potential energy and potential - electric potential energy - electric potential - calculating the potential from the field - potential due to point charges - potential due to continuous charge distributions - calculating the field from the potential - equipotential surfaces - the potential of a charged conductor chapter 29 The electric properties of materials - types of materials - a conductor in an electric field - ohmic materials - Ohm's law - an insulator in an electric field chapter 30 Capacitance - capacitors - capacitance chapter 31 DC circuits - electric current - electromotive force chapter 32 The magnetic field - the magnetic force on a moving charge - circulating charges - the Hall effect

3. QUANTUM MECHANICS reference text: J. J. Sakurai Modern Quantum Mechanics, Addison-Wesley chapter 1 Fundamental concepts - kets, bras, operators - base kets and matrix representation - measurements, observables, and uncertainty relations - position, momentum, and translation - wave functions in position and momentum space chapter 2 Quantum dynamics - time evolution and the Schroedinger equation - the Schroedinger versus the Heisenberg picture - simple harmonic oscillator - Schroedinger's wave equation chapter 3 Theory of angular momentum - rotations and angular momentum commutation relations - spin 1/2 systems and finite rotations - eigenvalues and eigenstates of angular momentum - orbital angular momentum - addition of angular momenta chapter 4 Symmetry in quantum mechanics - symmetries, conservation laws, and degeneracies - discrete symmetries, parity, or space inversion - lattice translation as a discrete symmetry - the time-reversal discrete symmetry chapter 5 Approximation methods - time independent perturbation theory: non degenerate case - time independent perturbation theory: the degenerate case

4. STATISTICAL MECHANICS reference text: K. Huang Statistical Mechanics, John Wiley & sons chapter 6 Classical statistical mechanics - the postulate of classical statistical mechanics - microcanonical ensemble - derivation of thermodynamics - equipartition theorem - classical ideal gas chapter 7 Canonical ensemble and grand canonical ensemble - canonical ensemble - energy fluctuations in the canonical ensemble - grand canonical ensemble - density fluctuations in the grand canonical ensemble - the chemical potential - equivalence of the canonical ensemble and grand canonical ensemble chapter 8 Quantum statistical mechanics - the postulate of quantum statistical mechanics - ensembles in quantum statistical mechanics - the ideal gases: micro canonical ensemble - the ideal gases: grand canonical ensemble chapter 11 Fermi systems - the equation of state of an ideal Fermi gas chapter 12 Bose systems - photons - Bose-Einstein condensation

5. ATOMIC AND MOLECULAR PHYSICS reference text: B. H Bransden & C. J. Joachain Physics of atoms and molecules, Longman Scientific & Technical chapter 3 One-electron atoms - the Schroedinger equation for one-electron atoms - energy levels - the eigenfunctions of the bound states chapter 6 Two-electron atoms - the Schroedinger equation for two-electron atoms - spin wave functions and the role of the Pauli exclusion principle - level scheme of two-electron atoms chapter 7 Many-electron atoms - the central field approximation - the periodic system of the elements chapter 9 Molecular structure - general nature of molecular structure - the Born-Oppenheimer separation for diatomic molecules - electronic structure of diatomic molecules - the structure of polyatomic molecules


Study modes

The course includes lectures on the theory (amounting to approximately 2/3 of the total number of hours dedicated to lecturing), alternated with tutoring sessions (amounting to approximately 1/3 of the total number of hours dedicated to lecturing), during which the methods to solve problems and exercises of the kinds that can be assigned in a written exam are treated.

Frequency modes: Attendance to the lectures is not mandatory but strongly recommended.



Exam mode

Written exam
There are two mid-term assessment tests during the course (lasting two hours each). If both tests are passed with a score of at least 15/30 and an average of not less than 18/30, the student is exempted from the written test for the entire academic year. The exemption expires at the end of the academic year to which it refers, namely September 2026. In case of failure, it is not possible to repeat a mid-term assessment test at a later time. The first mid-term assessment test concerns the lattice and electronic properties of solids. The second mid-term assessment test concerns the semiconductors and the vibrational properties of solid. Each test consists of two exercises each comprising various questions

There are 5 complete calls (written and oral): two in the January/February session, two in the June/July session and one in the September session.

The written test (lasting three hours) includes two problems, each one divided into several questions. The written test is passed with a score of no less than 18/30 and it is only valid for the session in which it was taken.  If a student decides to try the written exam in order to improve the grades obtained with the mid-term assessments, it is understood that the new grades overrule the previous ones, independently of the result of the written exam. Of course, the students have the right not to hand in their classwork if they feel unsure about their performance. In such a case, the previous grades are maintained.


Oral exam
The oral exam consists of an interview on the most relevant topics presented in the course. To pass the exam, the student must be able to present arguments and repeat calculations discussed and explained during the course. The student will be asked to apply the methods learned during the course to exercises or to examples and situations similar to those that were discussed in the course.
The evaluation takes into account:
- Correctness and completeness of the concepts discussed by the student;
- clarity and rigor of presentation;
- analytical development of the theory;
- problem-solving skills (method and results).

The final exam grade is determined by the average between the written score (or the average of the mid-term assessment tests) and the oral test score.

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PROGRAM
- Crystal structures, Bravais lattices [AM ch.4] -Reciprocal lattice  [AM ch. 5]
- Diffraction from crystals, structure factor [AM Ch. 6]
- Electrons in solids, Bloch theorem - Electronic bands - Nearly-free electrons - The tight-binding method  -  [AM ch. 8-10]. The concepts of holes and effective mass.
- Electrons in metals and interaction with the electromagnetic field (Dielectric function, transport properties of metals): Drude model and Sommerfeld model [AM ch. 1,2]. Semiclassical model [AM Ch. 12].
-Born-Oppenheimer approximation [notes]- Lattice vibrations, phonons -Specific heat (Einstein and Debye models, density of states) [AM-Ch. 22 p.421-443 and ch. 23]
- Intrinsic and extrinsic (doped) semiconductors - T dependence of the number of charge carriers [AM cap. 28]
- The Berry phase and the Anomalous Hall Effect [M. Grilli's notes]

Optional topics:
Boltzmann equation (relaxation-time approx.)  [Ziman, Sec. 7.1,7.2]
Physics of the p-n junction and applications to devices.
[AM ch. 29 p. 589-600].

Useful topics: AM  B, C, D, E, F, L Appendices.

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References

FUNDAMENTAL
- [AM] N.W. Ashcroft, N. D, Mermin, `Solid State Physics`, Holt-Saunders Int. Ed. 1981.
- [MG notes] Prof. Grilli's notes on Berry phase and Anomalous Hall Effect

ADDITIONAL

- J.M. Ziman, `Principles of the Theory of Solids', Cambridge University Press (1979)

- [BG] F. Bassani e U. M. Grassano, FISICA DELLO STATO SOLIDO, Bollati Boringhieri

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LECTURE DIARY 2025/26
AM= Ashcroft & Mermin, Solid State Physics

2025.10.01		General remarks on the CMP course. Bravais lattices Ch. 4 [AM}
2025.10.02		Direct lattice and reciprocal lattice Ch 4 and 5 [AM]
2025.10.08		The Reciprocal Lattice, X-ray diffraction experiments Ch. 5,6 [AM]
2025.10.09		X-ray diffraction experiments Ch. 6 [AM]
2025.10.10		Exercise on X-ray diffraction [mid-term test 2018.11.19 see online collection]
2025.10.15		Exercise on X-ray diffraction [mid-term test 2018.11.19) Structure factors and atomic form factora
2025.10.16		Periodic Boundary conditions and the Bloch Theorem (1st proof ch. 8 A-M)
2025.10.17		Exercise on X-ray diffraction [mid-term test 2021.12.17)
2025.10.22		Bloch Theorem (2nd proof ch. 8 A-M)
2025.10.23		Bloch Theorem, periodicity of el states in Reciprocal lattice, velocity of Bloch els.(ch 8)
2025.10.24		Exercise on X-ray diffraction [mid-term test 2021.12.17)
2025.10.29		Density of states, Nearly-Free Electrons: non-degenerate states
2025.10.30		Nearly-Free electrons: degenerate states
2025.10.31		Exercise on X-ray diffraction [mid-term test 2024.12.06)
2025.11.05		The Tight-Binding method (Ch. 10 AM)
2025.11.06		The Tight-Binding method (Ch. 10 AM); the sign of the hopping integrals, exercises
2025.11.07		Exercise on TBA for lattice without basis.
2025.11.12		Transport in metals: Drude model (A-M ch 1), heat conductivity. (A-M ch 1)
2025.11.13		Failures of the Drude model, quick summary of Sommerfeld model  (A-M ch 1-2), the semiclassical model
2025.11.14		The semiclassical model: effective mass, electrons and holes  (A-M ch 12)
2025.11.19		Summary on the semiclassical model (A-M ch 12).  Exercise on tigh-binding  with 2 atoms per unit cell
2025.11.20		The Born-Oppenheimer approximation (notes uploaded on this page)
2025.11.21		Exercise on TBA for 1D lattice with basis (~cuprate).
2025.11.26		Classical theory of 1D monoatomic harmonic crystal (A-M ch. 22)
2025.11.27		Classical theory of 1D biatomic harmonic crystal (A-M ch. 22)
2025.11.28		Exercise on tight-binding model and DOS (exam 12/02/2024 ex. 2)
2025.12.03		Classical theory of 3D monoatomic harmonic crystal (properties of the dynamical matrix) (A-M ch. 22)
2025.12.04		Classical theory of 3D monoatomic harmonic crystal. Quantization of the harmonic lattice (from L-C, relevant pages downloadable here)
2025.12.05		Exercises on DOS and question time
2025.12.10		Comments on Mid-Term Test Lattice specific heat at high and low temperature (A-M ch 23)
2025.12.11		Debye theory of lattice specific heat at intermediate temperature. Debye wavevector. (A-M ch 23)
2025.12.12		Debye frequency and temperature (A-M ch 23). Exs. on Debye theory of specific heat.
2025.12.17		Exercises on phonons and lattice specific heat.
2025.12.18		Intrinsic semiconductors (A-M ch 28)
2025.12.19		Extrinsic semiconductors. Transport in semiconductors (A-M ch 28)
2026.01.07		Intrinsic and extrinsic Semiconductors (A-M ch 28)
2026.01.08		Impurity levels in extr. semic. (A-M ch 28)
2026.01.09		exercises on phonons and on  intr. and extr. semic.
2026.01.14		Introduction to the Berry phase. [MG-notes]
2026.01.15		The Berry curvature in Condensed Matter: The Anomalous Hall effect [MG-notes]
2026.01.16		Mid-term test (phonons and semiconductors)